Answer :
To test for independence between the quality of management and the reputation of the company, we need to perform a Chi-Square test of independence. Below are the detailed steps involved in solving this problem.
### Given Data
We are provided with the following contingency table:
| Quality of Management | Excellent | Good | Fair |
|-----------------------|-----------|------|------|
| Excellent | 40 | 25 | 5 |
| Good | 35 | 35 | 10 |
| Fair | 25 | 10 | 15 |
### Step-by-Step Solution
1. Formulate the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The quality of management and the reputation of the company are independent.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The quality of management and the reputation of the company are not independent.
2. Calculate Expected Frequencies:
The expected frequency for each cell in a contingency table is calculated using the formula:
[tex]\[ E_{ij} = \frac{(Row\ Total)_{i} \times (Column\ Total)_{j}}{Grand\ Total} \][/tex]
Given the observed frequency table:
| Quality of Management | Excellent | Good | Fair | Row Totals |
|-----------------------|-----------|-------|-------|------------|
| Excellent | 40 | 25 | 5 | 70 |
| Good | 35 | 35 | 10 | 80 |
| Fair | 25 | 10 | 15 | 50 |
| Column Totals | 100 | 70 | 30 | 200 |
We can calculate expected frequencies for each cell. However, the result from Python calculation confirms that the observed {Classical Calculations} matches the calculated expected frequencies.
3. Calculate the Chi-Squared Test Statistic:
The Chi-Squared ([tex]\( \chi^2 \)[/tex]) statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
where [tex]\(O_{ij}\)[/tex] represents observed frequencies, and [tex]\(E_{ij}\)[/tex] represents expected frequencies.
The calculated value of [tex]\(\chi^2\)[/tex] from previous computations is:
[tex]\[ \chi^2 = 17.03 \][/tex]
4. Determine the Degrees of Freedom:
The degrees of freedom ([tex]\(df\)[/tex]) for a contingency table is given by:
[tex]\[ df = (r - 1) \times (c - 1) \][/tex]
where [tex]\(r\)[/tex] is the number of rows and [tex]\(c\)[/tex] is the number of columns.
For our table:
[tex]\[ df = (3 - 1) \times (3 - 1) = 2 \times 2 = 4 \][/tex]
5. Find the [tex]\(p\)[/tex]-Value:
Using the Chi-Square distribution table or computational tools, we find the [tex]\(p\)[/tex]-value corresponding to [tex]\(\chi^2 = 17.03\)[/tex] with [tex]\(df = 4\)[/tex].
The calculated [tex]\(p\)[/tex]-value is [tex]\(0.00\)[/tex].
6. Conclusion:
Compare the [tex]\(p\)[/tex]-value to the significance level ([tex]\(\alpha = 0.05\)[/tex]):
[tex]\[ p\ \text{value} = 0.00 < 0.05 = \alpha \][/tex]
Since the [tex]\(p\)[/tex]-value is less than the significance level, we reject the null hypothesis.
### Conclusion & Interpretation
We conclude that there is sufficient evidence to suggest that the quality of management is not independent of the reputation of the company at the [tex]\( \alpha = 0.05 \)[/tex] level of significance. This suggests a relationship between the ratings for quality of management and the reputation of the company. Hence, the rating for the quality of management is dependent on the rating for the reputation of the company. The highest probabilities reflect a notable association between these two variables.
### Given Data
We are provided with the following contingency table:
| Quality of Management | Excellent | Good | Fair |
|-----------------------|-----------|------|------|
| Excellent | 40 | 25 | 5 |
| Good | 35 | 35 | 10 |
| Fair | 25 | 10 | 15 |
### Step-by-Step Solution
1. Formulate the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The quality of management and the reputation of the company are independent.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The quality of management and the reputation of the company are not independent.
2. Calculate Expected Frequencies:
The expected frequency for each cell in a contingency table is calculated using the formula:
[tex]\[ E_{ij} = \frac{(Row\ Total)_{i} \times (Column\ Total)_{j}}{Grand\ Total} \][/tex]
Given the observed frequency table:
| Quality of Management | Excellent | Good | Fair | Row Totals |
|-----------------------|-----------|-------|-------|------------|
| Excellent | 40 | 25 | 5 | 70 |
| Good | 35 | 35 | 10 | 80 |
| Fair | 25 | 10 | 15 | 50 |
| Column Totals | 100 | 70 | 30 | 200 |
We can calculate expected frequencies for each cell. However, the result from Python calculation confirms that the observed {Classical Calculations} matches the calculated expected frequencies.
3. Calculate the Chi-Squared Test Statistic:
The Chi-Squared ([tex]\( \chi^2 \)[/tex]) statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
where [tex]\(O_{ij}\)[/tex] represents observed frequencies, and [tex]\(E_{ij}\)[/tex] represents expected frequencies.
The calculated value of [tex]\(\chi^2\)[/tex] from previous computations is:
[tex]\[ \chi^2 = 17.03 \][/tex]
4. Determine the Degrees of Freedom:
The degrees of freedom ([tex]\(df\)[/tex]) for a contingency table is given by:
[tex]\[ df = (r - 1) \times (c - 1) \][/tex]
where [tex]\(r\)[/tex] is the number of rows and [tex]\(c\)[/tex] is the number of columns.
For our table:
[tex]\[ df = (3 - 1) \times (3 - 1) = 2 \times 2 = 4 \][/tex]
5. Find the [tex]\(p\)[/tex]-Value:
Using the Chi-Square distribution table or computational tools, we find the [tex]\(p\)[/tex]-value corresponding to [tex]\(\chi^2 = 17.03\)[/tex] with [tex]\(df = 4\)[/tex].
The calculated [tex]\(p\)[/tex]-value is [tex]\(0.00\)[/tex].
6. Conclusion:
Compare the [tex]\(p\)[/tex]-value to the significance level ([tex]\(\alpha = 0.05\)[/tex]):
[tex]\[ p\ \text{value} = 0.00 < 0.05 = \alpha \][/tex]
Since the [tex]\(p\)[/tex]-value is less than the significance level, we reject the null hypothesis.
### Conclusion & Interpretation
We conclude that there is sufficient evidence to suggest that the quality of management is not independent of the reputation of the company at the [tex]\( \alpha = 0.05 \)[/tex] level of significance. This suggests a relationship between the ratings for quality of management and the reputation of the company. Hence, the rating for the quality of management is dependent on the rating for the reputation of the company. The highest probabilities reflect a notable association between these two variables.
